The Assault class in XCOM: Enemy Unknown does just as the title implies; it is designed for intensive assaults against heavily fortified enemies. As such, it will be the unit both dealing the most damage and taking the most damage. You will want to make sure that you have the underlying support system necessary to keep your Assault guys alive. 

Assault is a great class to double up on in your party because of this duality.

The following is the list of skills available to Assault classes. Note that you can only choose one ability per rank up.

Squaddie: 

Run & Gun: Allows firing or Overwatch after Dashing. Two turn cooldown.

Corporal:

Tactical Sense: Confers +5 Defense per enemy in sight (man +20)

Aggression: Confers +10% critical chance per enemy in sight (max +30%)

Sergeant:

Lightning Reflexes: Forces the first reaction shot against this unit each turn to miss.

Close and Personal: Confers +30% Critical Chance against adjacent targets - bonus declines with distance from target.

Lieutenant:

Flush: Fire a shot that causes enemies to run out of cover.

Rapid Fire: Take two shots against a single target in quick succession. Each shot carries a -15 penalty to aim.

Captain:

Close Combat Specialist: Confers a reaction shot against any enemy who closes within 4 tiles.

Bring 'Em On: Adds 1 damage on critical hits for each enemy the squad can see (max 5)

Major:

Extra Conditioning: Confers bonus health based on which type of armor is equipped. Heavier armor increases the bonus.

Colonel:

Resilience: Confers immunity to critical hits.

Killer Instinct: Activating Run & Gun now also grants +50% critical damage for the rest of the turn.

So here are the strategies for the Assault class. You should probably have two Assault guys in your party at any given time, allowing you to spec one to offense and one to defense. For the defense guy, try Tactical Sense, Lightning Reflexes, Flush, Close Combat Specialist, Resilience. For the offense guy, spec him out with Aggression, Close and Personal, Rapid Fire, Close Combat Specialist, and Killer Instinct.